Extension d’une valuation

Abstract:

We want to determine all the extensions of a valuation $\nu$ of a field $K$ to a cyclic extension $L$ of $K$, i.e. $L=K(x)$ is the field of rational functions of $x$ or $L=K(\theta )$ is the finite separable extension generated by a root $\theta$ of an irreducible polynomial $G(x)$. In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation $\mu$ of $K[x]$, and has shown how we can recover any extension to $L$ of a discrete rank one valuation $\nu$ of $K$ by a countable sequence of augmented valuations $\bigl (\mu _i\bigr ) _{i \in I}$, with $I \subset \mathbb N$. The valuation $\mu _i$ is defined by induction from the valuation $\mu _{i-1}$, from a key polynomial $\phi _i$ and from the value $\gamma _i = \mu ( \phi _i )$. In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation $\nu$ of $K$. For this we need to introduce simple admissible families of augmented valuations ${\mathcal A} = \bigl ( \mu _{\alpha } \bigr ) _{\alpha \in A}$, where $A$ is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then, any extension $\mu$ to $L$ of a valuation $\nu$ on $K$ is again a limit of a family of augmented valuations. We also get a “factorization” theorem which gives a description of the values $( \mu _{\alpha } (f))$ for any polynomial $f$ in $K[x]$.